Superconductivity from energy fluctuations in dilute quantum critical polar metals

Superconductivity in low carrier density metals challenges the conventional electron-phonon theory due to the absence of retardation required to overcome Coulomb repulsion. Here we demonstrate that pairing mediated by energy fluctuations, ubiquitously present close to continuous phase transitions, occurs in dilute quantum critical polar metals and results in a dome-like dependence of the superconducting Tc on carrier density, characteristic of non-BCS superconductors. In quantum critical polar metals, the Coulomb repulsion is heavily screened, while the critical transverse optical phonons decouple from the electron charge. In the resulting vacuum, long-range attractive interactions emerge from the energy fluctuations of the critical phonons, resembling the gravitational interactions of a chargeless dark matter universe. Our estimates show that this mechanism may explain the critical temperatures observed in doped SrTiO3. We provide predictions for the enhancement of superconductivity near polar quantum criticality in two- and three-dimensional materials that can be used to test our theory.

(which are transverse). Therefore the next available coupling is: H En = g dx 3 ρ e (r) (P (r)) 2 (1) where ρ e (r) is the charge density operator and P (r) is the local polarization. The fluctuation of the charge density around its equilibrium value δρ e (r) = ρ e (r) − n e can lead to an effective attractive interaction in the perturbation expansion (the first term is proportional to g 2 ): V En (iω n , q) = − 2π 2 α Reviewer #1 (Remarks to the author): The authors built a low-energy effective model in an attempt to study dilute quantum critical polar metals. The motivation is that in dilute limit, the Fermi surface is small, which constrains the phase space and leads to negligible direct coupling between itinerant electrons and polar phonon modes This attractive interaction has the origin of two-phonon exchange. At low carrier density, this attractive interaction will overcome Coulomb repulsion and lead to possible superconductivity. An effective electron-phonon coupling λ is: which has dome-like dependence on k F . Since k F ∼ n 1/3 e and the superconducting transition temperature T c ∼ E F exp(−1/λ). Therefore, T c has a dome shape as a function of carrier density n e . The authors applied this theory to doped SrTiO 3 . Eq.
(1) has been studied in literature (e.g. PRL 126, 076601 (2021)) and the resulting attractive interaction that arises from two-phonon exchange has been proposed as possible mechanism to explain superconductivity of doped SrTiO 3 (e.g. PRL 32, 215 (1974) and PR Research 1, 013003 (2019)). The novelty in this study is that the authors find that the charge fluctuation in Eq. (1) can lead to this attractive interaction and thus justify the two-phonon exchange as the pairing mechanism for superconductivity in dilute metals.
However, I have a few comments/questions for the authors to address: 1. The theory is supposed to describe "dilute quantum critical polar metals". What parameter is tuned in order to achieve the polar quantum critical point (QCP)? In experiments of Ca x Sr 1−x TiO 3−δ (NP 13, 643 (2017)) and doped BaTiO 3 (PRL 104, 147602 (2010)), doped electrons suppress the polar displacements and lead to a polar QCP at a finite carrier density n e . However, in the current theory, the transverse polar phonon mode has dependence on n e as: ω T (n e ) = ω 2 T 0 + gn e 0 Ω 2 0 (4) where ω T 0 > 0 is the transverse polar phonon frequency when mobile carriers vanish. From Eq. (4), does that mean g < 0 so as to achieve a polar QCP at a finite n e ? However, if g < 0, does that imply Eq. (1) may lead to some instability when |P (r)| is sufficiently large? If g > 0, then where is the polar QCP? I suppose that ω T 0 > 0 is a constant throughout the calculations. Clarification is needed here. 2. In Fig. 2, the authors show that the effective pairing potential V P air En may have three different forms as the carrier density n e changes. For low carrier density (middle region), For high carrier density (rightmost region), For intermediate density (leftmost region), V P air En (r, τ ) ∼ δ(r)δ(τ ) log(ξa 0 ) There are a few confusing points that need to be clarified: * the x-axis of Fig. 2 is monotonic with the carrier density n e . Then why does the leftmost region correspond to the "intermediate density"? * the paper wrote "For a low density polar metal that is critical at zero doping ω T (n e = 0) = 0", that means ω T 0 = 0. Is that a necessary condition to derive Eq. (5)? What happens if ω T 0 > 0? * For a finite ω T 0 > 0, since 2c s k F ∼ n 1/3 e and E F ∼ n 2/3 e , when n e is sufficiently small, ω T → ω T 0 , which must exceed 2c s k F and E F . Then does Eq. (7) correspond to ultralow densities, rather than "intermediate density"? 3. When the authors apply their theory to doped SrTiO 3 , can they estimate what the two critical carrier concentrations are in Fig. 2, given the experimental information (e.g. c s , 1 )? If it is doped BaTiO 3 , whether these two critical carrier concentrations will be very different?
(1) is allowed by symmetry, what is the microscopic origin of this term? In particular, what does g depend on? Is it strongly materialdependent? For example, in doped SrTiO 3 , the authors use g/a 3 0 = 0.62. Then how about doped BaTiO 3 or other doped ferroelectric semiconductors? A more general question is: what is the condition under which Eq. (1) becomes the dominant interaction in addition to Coulomb repulsion? Is it that the material must be close to the polar QCP? Or is it that the carrier density must be low? Or both? 5. The presentation needs a substantial improvement. This is a pure theoretical modelling. However, the notations are sometimes confusing. An incomplete list include: * In the main text, Eq. (11), (13), (18), what does iω mean? Is ω a real frequency or a Matsubara frequency? Eq. (17) uses iω n . Do they mean the same thing? * In the main text, Eq. (16) n max has the dimension 1/[length] 3 . Then what does the text n max 1 mean? * In the main text, we have n e and n. Do they mean the same thing (i.e. carrier density)? If yes, can we just use one symbol to avoid confusion? * In the main text, we have ω T (q), ω T (n e ) and a constant ω T . In the SI, we have ω T (n e , q). In the derivation, we have ω T ω n . What does that ω T mean? It means the constant term (i.e. n e = 0 and q = 0), or a more general ω T (n e , q)? * In the SI, Eq. (1) reads S = S e + S ph + S e−ph . However, S e−ph is not defined at all. Why not reads S = S e + S ph + S En + S Coul explicitly? * Sometimes a vector is denoted as an arrow on top of a letter and in other occasions is denoted as bond font. Is there a particular reason for this? 6. Finally I have a comment: the so-called "dilute quantum critical polar metal" is very rare. Probably doped SrTiO 3 is the only known example. SrTiO 3 is unique is that the material by itself is in the vicinity of ferroelectric phase transition. In other polar metals (such as LiOsO 3 ) or doped ferroelectric materials (such as doped BaTiO 3 ), while it is possible to drive them close to a polar QCP via strain or doping, usually the carrier density around the polar QCP is reasonably high (∼ 10 21 /cm 3 ) and therefore the Fermi surface is sufficiently large so that the conventional electron-phonon mechanism is working. Therefore I am wondering whether the current pairing mechanism can be applied to any material system other than doped SrTiO 3 ?
1. The first important step in your study is the calculation of the effective electronelectron attraction mediated by the two-phonon processes at different dopings. Depending on the doping and the proximity to the critical point, the behavior is very different. Could you summarize all the results for different regimes in a concise readerfriendly manner? I believe Fig. 2 is supposed to help with that but I find it hard to understand its meaning properly. For example, what are the densities that correspond to the vertical grey dashed lines on that figures? Is there a regime corresponding to Eq. (10)? When you compare omega_T to c*k_F, do you have in mind the bare TO phonon mass or the one renormalized according to Eq. (6)? What's the meaning of different curves in Fig. 2? 2. You have introduced too many similar notations having different meaning. It makes it really hard to navigate across the paper. For example, there's \omega_T, \omega_T(n_e), \omega_T(q), \omega_q, \Omega_T, Omega_0 all meaning different things.
3. For some reason, you spend a lot of time discussing the analogy with the "dark matter". Of course, we all love analogies between different concepts in physics, but this one seems a bit inappropriate. I'm not an expert in "dark matter", but I don't think people already know convincingly what that is and how exactly it interacts with more conventional matter. To me, the problem of the two-phonon mediated superconductivity is interesting enough on its own, just from the condensed matter perspective. Moreover, sentences like "We find that polar quantum criticality results in long-range "gravitational" interactions that mediate attraction between the electrons in an electromagnetically neutral background." almost made me believe that the nature of this interaction is gravitational rather than electromagnetic, which is extremely confusing.
4. Instead of discussing the relation to the "dark matter", I'd suggest to discuss the similarities and differences with some other works on the same topic. Specifically, it'd be very interesting to see some comments regarding this recent preprint: https://arxiv.org/abs/2106.09530. The authors of that work also obtained some log enhancement of the effective interaction in the Cooper channel, see their Eqs. (7)-(8), which, however, seems different from your log in Eq. (14), since it doesn't depend on k_F (on electron density). I understand that this paper appeared after yours, but nevertheless I'd be happy to see your comments. 5. You state above Eq. (7) that the Fermi liquid state is not destroyed even at the QCP based on the observation that the two-phonon coupling is irrelevant. But is it a sufficient condition? Can it happen that some loop diagrams lead to some crucial (possibly nonanalytic) corrections which eventually destroy the Fermi liquid at the QCP (like, e.g., Landau damping in a more conventional coupling to a critical mode)? I'm not encouraging you to perform such an analysis in this paper, but rather curious whether this statement is sufficiently substantiated.
In addition, I have some minor more technical questions and comments: 6. The second paragraph of the introduction mentions the alternative mechanism for coupling to the TO mode through the spin-orbit coupling. Then, the last sentence of this paragraph , "However, this appealing idea encounters a difficulty, for the critical modes of a polar QCP are transverse optic (TO) phonons that decouple from the electrons at low momenta", looks a bit strange and confusing since it is exactly the spin-orbit coupling that allows to couple electrons to the TO phonons directly even at zero momentum. Also, it would be appropriate to mention here some earlier papers that discussed that type of coupling and its consequences for superconductivity: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.74.012501, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.115.207002, https://journals.aps.org/prx/abstract/10.1103/PhysRevX.9.031046 7. The first paragraph of the intro says "A challenge to this mechanism is posed by superconductivity in low carrier metals near polar quantum critical points (QCPs)." I believe it's meant to be "... in low carrier density metals..." 8. Above Eq. (2), the term S_e should contain the sum over k, not just \vec{k} (i.e., sum over frequencies is missing). 9. In Eq. (3), the last term should read \delta_{\alpha \beta} -\hat q_{\alpha} \hat q_{\beta}, not just 1 -\hat q_{\alpha} \hat q_{\beta} (i.e., 1 should be substituted with \delta_{\alpha \beta}). 18. What \omega_T do you use to plot Fig. 3? Again, are these the bare TO frequencies or the ones renormalized at finite electron density according to Eq. (6)? If these are the bare frequencies (masses), what happens to the effective interaction and to superconducting T_c right at the "real" QCP at finite density, i.e., when \omega_T(n_e)=0? What's the form of the effective interaction in this case?
19. The bare electron's action presented in the supplement seems to disagree with the one from the main text. There's an extra minus sign (I believe the one in the main text is the correct one).

Response to the Referees
We thank both the referees for their careful reading of our manuscript, and for their many comments on the science therein. We are particularly appreciative of your detailed suggestions that have helped to clarify our presentation. Referee 1 acknowledges "the novelty of this study," noting that we 'justify the two-phonon exchange as the pairing mechanism for superconductivity in dilute metals." Referee 2 writes that "The study is timely, while the topic and the results are very interesting." Because of the length of our detailed responses to the Reviewers, here we provide a summary of the key points of our work: • In dilute quantum critical polar metals, the Coulomb repulsion is heavily screened and the soft polar mode decouples from the charge in the case of negligble spin-orbit effects (which we will assume).
• Experimentally the superconducting transition temperature is enhanced with proximity to the polar quantum critical point. Since the electrons do not interact directly with the polar modes, we propose that it is the energy density of the zero-point polar fluctuations that drives the electron-electron attraction.
• We find that this electron-electron attraction, mediated by energy fluctuations, overcomes the Coulomb repulsion at low charge densities close to the polar quantum critical point resulting in a superconducting instability • Application of our ideas to doped SrTiO 3 (STO) shows good agreement with experiment in the superconducting regime with the appropriate density. Since this interaction is known to describe anomalous electron conduction in the normal state, our theory provides a unified approach to transport in two different temperature regimes of doped STO; furthermore it explains the observed phonon frequency shift with doping.
• We predict that this mechanism will be considerably enhanced in two-dimensional quantum critical polar systems that can be realized in epitaxial films.
Please find below our itemized replies to the Reviewers' specific points. Please note that,

Referee 1
The novelty in this study is that the authors find that the charge fluctuation in (1) can lead to this attractive interaction and thus justify the two-phonon exchange as the pairing mechanism for superconductivity in dilute metals.

Comment:
We would like to clarify that the pairing in our approach is due to quantum critical fluctuations of the energy density of the local polarization [( P (r)) 2 in (1)]. While perturbatively the resulting attraction can be attributed to two-phonon exchange, phonon nonlinearities will lead to the energy fluctuation exchange being different from two-phonon one. For d = 3 the corrections are logarithmic but, as we discuss, in d = 2 the quartic interactions between the critical phonons are relevant and must be included.
However, I have a few comments/questions for the authors to address: 1. The theory is supposed to describe "dilute quantum critical polar metals". (2) where ω T 0 > 0 is the transverse polar phonon frequency when mobile electrons vanish. From Eq.
(2), does that mean g < 0 so as to achieve a polar QCP at a finite n e ? However, if g < 0, then where is the polar QCP? I suppose that ω T o > 0 is a constant throughout the calculation. Clarification is needed here.
Reply: We certainly agree that in Ca x Sr 1−x TiO 3 and in doped BaTiO 3 , the polar mode is observed to harden with electron density; similar effects have been observed in SrTiO 3 . It is for this reason that we take the coupling constant g to be positive in (2). Here ω 2 T 0 refers to the undoped case; it is a bare quantity in the Lagrangian that measures the curvature of the free energy with respect to the transverse polarization. Different curves in Fig. 3 correspond to different values of ω 2 T 0 ≥ 0. However we assume that at all the densities considered the transverse optical phonon energy ω T (n e ) is small with respect to Ω 0 and to Ω T , leading to respectively a large lattice dielectric constant and to a large logarithmic enhancement of the attraction in Eq. (11) of the main text; this enhanced attraction originates from quantum fluctuations of the phonon energy density and therefore our theory does indeed include quantum critical phenomena.
If ω 2 T 0 < 0 in the bare Lagrangian, the system is polar and subsequent doping leads to This is indeed the situation for the two materials that you mention, Ca x Sr 1−x TiO 3−δ and doped BaTiO 3 ; the insulating systems are initially polar (ω 2 T 0 < 0), and there is a true quantum phase transition at finite doping (see Nat. Phys 13, 643 (2017)).
As mentioned briefly in the text, it would be interesting to generalize our theory to the ordered (polar) phase. On general grounds, order parameter and energy fluctuations are also expected to be prominent in the ordered phase close to the transition.
In the course of our revision, we found that a factor of two had to be included in Eq. (6) of the main text; this equation and its results displayed in Figure 3 have now been updated. We found no appreciable difference for the lower density range of our focus. However the results are somewhat different at higher densities, where the Matsubara frequency-dependence of both the screened Coulomb repulsion and the energy fluctuation-mediated attraction have to be included. A discussion of this point has been added on page six of the revised manuscript.
Action: We have addressed and clarified these points in the revised text -see the discussion after Eq. 6 in the main text and in the middle of the left-hand column on page 6. Fig. 2, the authors show that the effective pairing potential V P air En may have three different forms as the carrier density n e changes. For low carrier density (middle region),

In
For high carrier density (rightmost region), For intermediate density (leftmost region), There are a few confusing points that need to be clarified: leftmost region correspond to the "intermediate density"?
Reply: We acknowledge the referee's point and thank him/her for pointing out our error. In the main text, we mislabelled the low-density regime as the intermediate doping one. We apologize for the resulting confusion and this point has been corrected.
2 (b): The paper wrote "For a low density polar metal that is crtical at zero doping ω T (n e = 0) = 0", that means ω T 0 = 0. Is that a necessary condition to derive Eq. (2)?
Reply: The necessary condition for the derivation of (2) is c s k F ω T 0 .
Action: We have added a clarification of this point after Eq. (12) in the main text.
e , when n e is sufficiently small, ω T → ω T 0 ., which must exceed 2c s k F and E F . Then does Eq. 3. When the authors apply their theory to doped SrTiO 3 , can they estimate what the two critical carrier concentrations are in Fig. 2, given the experimental information (e.g. c s , 1 )?
If it is doped BaTiO 3 , whether these two critical carrier concentrations will be very different?
Reply: The densities in Figure 2 describe crossovers between different interaction regimes of the quantum critical polar metal. For STO, we estimate that the lower and higher density crossovers occur around n 1 ∼ 1.3 × 10 17 and n 2 ∼ 2.6 × 10 19 respectively; note that we have modeled the "intermediate density" region when fitting to the three-dimensional STO data in Figure 3a. These distinct densities depend on several parameters that are material-specific; in particular, n 1 ≈ 1 3π 2 ω T 0 2cs 3 , n 2 = 3π 2 4m * cs 3 .
In the absence of doping, at ambient pressure BaTiO 3 has a high temperature ferroelec-tric transition so that ω 2 T 0 < 0. Suppression of the polar transition temperature with doping has been reported, with T C going to zero at n * ≈ 1.9 · 10 21 cm −3 [PRL 104, 147602 (2010)].
Taking the effective mass in BaTiO 3 to be around 10 electron masses [PRB 78, 045107 (2008)] and c s around 1.5 times larger than in SrTiO 3 [c 2 s = 4750 (mevÅ) 2 from PRB 4, 155 (1971)], we find that n 2 ≈ 1.5 · 10 21 cm −3 . Therefore, at the QCP this system is already in the right-hand region of Fig. 2  the dominant interaction in addition to Coulomb repulsion? Is it that the material must be close to the polar QCP? Or is it that the carrier density must be low? Or both?
Reply: In short we need both conditions: proximity to the QCP so that the electronelectron interactions are heavily screened, and low density so that the attraction due to energy fluctuations is large enough to overcome the Coulomb repulsion. In particular, Eq.
(13) of the main text implies suppression of hte attractive interaction with increase in the density-dependent quantitites k f and ω T ; the latter set the scale of the momentum transfer and the distance from the QCP respectively. Reply: We thank the referee for pointing out this oversight. Here it should have been n max a 3 0 and this has been corrected.

(c):
In the main text, we have n e and n. Do they mean the same thing (i.e. carrier density)? If yes, can we just use one symbol to avoid confusion?
Reply: Yes, they are indeed both the electron density; we have replaced n with n e throughout the text.

(d)
In the main text, we have ω T (q), ω t (n e ), and a constant ω T . In the SI, we have ω(n e , q). In the derivation, we have ω T ω n . What does that ω T mean? It means the constant term (i.e. n e = 0 and q = 0), or a more general ω T (n e , q)?
Reply: Again we apologize for the ambiguous notation and we have tidied it up in the revised text. In particular, for each quantity we have indicated its dependent variables.
Thus the optical phonon energy ω T (n e , q) depends on both the electron density n e and momentum q. ω T 0 ≡ ω T (n e = 0) is the transverse phonon energy at q = 0 and n e = 0.
Action: We have made the notation more transparent and consistent in the revised text and the Supplementary Material.

(f )
In the SI, Eq.
(1) reads S = S e + S ph + S e−ph . However, S e−ph is not defined at all.
Why not read S = S e + S ph + S En + S Coul explicitly?
Reply: We thank the Referee for pointing out this inconsistency -in fact, this is a typo, which we have corrected in the new version. Indeed S En is meant instead of S e−ph .

(g) Sometimes a vector is denoted as an arrow on top of a letter and in other occasions
is denoted as bold font. Is there a particular reason for this?
Reply: We have unified the vector notation in the revised text.
Action: We have streamlined our notation throughout the revised main text and in the Supplementary Information to address the points from this set of questions.
6. Finally I have a comment: the so-called "dilute quantum critical polar metal" is very rare.
Probably doped SrTiO 3 is the only known example. SrTiO 3 is unique in that the material by itself is in the vicinity of ferroelectric phase transition. In other polar metals (such as LiOsO 3 or doped ferroelectric materials (such as doped BaTiO 3 ), while it is possible to drive them close to a polar QCP via strain or doping, usually the carrier density around the polar QCP is reasonably high ( 10 21 /cm 3 ) and therefore the Fermi surface is sufficiently large so that the conventional electron-phonon mechanism is working. Therfore I am wondering whether the current pairing mechanism can be applied to any material system other than doped SrTiO 3 ?
Reply: We thank the Referee for the opportunity to discuss this important point. There are good reasons to believe that our theory applies to several polar metals and thus not just to STO; these include other doped perovskites such as BaTiO 3 and KTaO 3 . In the case of

Referee 2
The study is timely, while the topic and the results are very interesting. Although I think the paper deserves publication in some form, I believe the presentation can be improved significantly. I'll try to list the most important aspects that bothered me: 1. The first important step in your study is the calculation of the effective electron-electron attraction mediated by the two-phonon processes at different dopings. Depending on the doping and the proximity to the critical point, the behavior is very different. Could you summarize all the results for different regimes in a concise reader-friendly manner? I believe Fig. 2 is supposed to help with that but I find it hard to understand its meaning properly.
For example, what are the densities that correspond to the vertical grey dashed lines on that figures? Is there a regime corresponding to Eq. (10)? When you compare ω T to c * k F , do you have in mind the bare TO phonon mass or the one renormalized according to Eq.
Reply: The curves in Fig. 2  At low momenta, one has ω 2 T (n e , q) ≈ ω 2 T (n e )+c 2 s q 2 . We ignore the effects of renormalization of c s with changing electron density, as these effects are absent to lowest order in the small coupling g (see Eq. (6) of the main text). ω T 0 ≡ ω T (n e = 0) is the transverse phonon energy at q = 0 and n e = 0. At sufficiently low densities (n e ω 2 T 0 2gε 0 Ω 2 0 ; left side of Fig. 2) ω T (n e ) ≈ ω T 0 . Finally, Ω T = max q ω T (n e , q) provides the large-energy cutoff for the phonon energies.
Action: We have streamlined and clarified the notation of the different frequencies we have introduced.
3. For some reason, you spend a lot of time discusssing the analogy with the "dark matter".
Of course, we all love analogies between different concepts in physics, but this one seems a bit inappropriate. I'm not an expert in "dark matter', but I don't think people already know convincingly what that is and how exactly it interacts with more conventional matter.
To me, the problem of the two-phonon mediated superconductivity is interesting enough on its own, just from the condensed matter perspective. Moreover, sentences like "We find that polar quantum criticality results in long-range "gravitational" interactions that mediate attraction between the electrons in an electromagnetically neutral background." almost made me believe that the nature of this interaction is gravitational rather than electromagnetic, which is extremely confusing.
Reply: The reason we chose this analogy is that in fact, dark matter is known to interact with baryons gravitationally (this is how it has been detected), i.e. via the stress-energy tensor. Furthermore, this is to highlight the universal character of interactions mediated by energy fluctuations, rather than fluctuations of, e.g. spin or charge. Of course, the ultimate origin of this phenomenon is electrostatic, but the long-wavelength, emergent physics is driven by fluctuations in the energy against a charge neutral background. We have deliberately made the distinction between energy fluctuation exchange and two-phonon exchange to highlight the potential differences due to nonlinearities and non-perturbative effects. In the d = 3 + 1 case, the corrections to the two-phonon exchange due to phonon nonlinearities are only logarithmically divergent, but in d = 2 + 1 as we have mentioned in the paper, the energy fluctuations pick up an anomalous dimension, and they are fully distinct from the perturbative two-phonon processes. In short, it is more accurate to characterize the mechanism as being that of energy fluctuations, rather than perturbative two-phonon processes. 4. Instead of discussing the relation to the "dark matter", I'd suggest to discuss the similarities and differences with some other works on the same topic. Specifically, it'd be very inter-esting to see some comments regarding this recent preprint: https://arxiv.org/abs/2106.09530.
The authors of that work also obtained some log enhancement of the effective interaction in the Cooper channel, see their Eqs. (7)-(8), which, however, seems different from your log in Eq. (14), since it doesn't depend on k F (on electron density). I understand that this paper appeared after yours, but nevertheless I'd be happy to see your comments.
Reply: The work mentioned by the Referee has two important distinctions from ours.
First, it concerns the regime of extremely low densities, corresponding to the region I in By contrast superconductivity at roughly 10 19 cm −3 , our density regime of focus, has been observed for all types of doping and does display a Meissner effect, indicating that it is a bulk phenomenon.
Action: We have noted this recent paper in the conclusion of our revised text.

5.
You state above Eq. (7) that the Fermi liquid state is not destroyed even at the QCP based on the observation that the two-phonon coupling is irrelevant. But is it a sufficient condition? Can it happen that some loop diagrams lead to some crucial (possibly nonanalytic) corrections which eventually destroy the Fermi liquid at the QCP (like, e.g., Landau damping in a more conventional coupling to a critical mode)? I'm not encouraging you to perform such an analysis in this paper, but rather curious whether this statement is sufficiently substantiated. 6.The second paragraph of the introduction mentions the alternative mechanism for coupling to the TO mode through the spin-orbit coupling. Then, the last sentence of this paragraph , "However, this appealing idea encounters a difficulty, for the critical modes of a polar QCP are transverse optic (TO) phonons that decouple from the electrons at low momenta", looks a bit strange and confusing since it is exactly the spin-orbit coupling that allows to couple electrons to the TO phonons directly even at zero momentum. Also, it would be appropriate to mention here some earlier papers that discussed that type of coupling and its consequences 7. The first paragraph of the intro says "A challenge to this mechanism is posed by superconductivity in low carrier metals near polar quantum critical points (QCPs)." I believe it's meant to be "... in low carrier density metals..." 8. Above Eq. (2), the term S e should contain the sum over k, not just k (i.e., sum over frequencies is missing).
Reply: We thank the Referee for spotting these errors and we have corrected them.
Action: We have addressed these points in the revised text.
10. Below Eq. (3), you present an expression of ω 2 L (q) which seems to be taken at q = 0, i.e., it looks like ω 2 L (0). pendently from energy fluctuation coupling and therefore can be important. In particular, in d = 2 + 1 they do affect the phonon Green's function (see Ref. 35) introducing an anomalous dimension. We use this to argue that the Fermi liquid is preserved and g is irrelevant in the main text, and we elaborate in the Supplementary Material. In d = 3 + 1 only weak logarithmic corrections are present due to u, which we neglect. We do assume g to be small so that self-energy effects due to g can be neglected provided that the coupling g is irrelevant (which we demonstrate to be true in the Supplementary Material).
Action: The discussion of quartic interactions in the 2D case has been clarified to address this point more extensively.

18.
What ω T do you use to plot Fig. 3? Again, are these the bare TO frequencies or the ones renormalized at finite electron density according to Eq. (6)? If these are the bare frequencies (masses), what happens to the effective interaction and to superconducting T c right at the "real" QCP at finite density, i.e., when ω T (n e ) = 0? What's the form of the effective interaction in this case?
Reply: We use ω T (n e ) given by Eq. (6) [i.e. renormalized ones], with the value of g given in the caption and ω T 0 indicated in the panels. Since g > 0 and ω 2 T 0 ≥ 0 , the system does not have a finite-density QCP. Generally, in case ω T (n e ) = 0, the interaction would be the one corresponding to regime II or III in Fig. 2, depending on the density. This can be deduced from Eq. (13) that is valid for all regimes.
Action: We have streamlined our frequency notation in the revised text and have also provided more discussion of Equation (6) (main text) 19. The bare electron's action presented in the supplement seems to disagree with the one from the main text. There's an extra minus sign (I believe the one in the main text is the correct one).

Reply to the Referees
We thank both Reviewers for their careful readings of our responses to their detailed comments. We are also grateful for their positive assessments of our revised manucript and for Reviewer 2's recommendation that it should be published. Here we address their requests for clarification of specific points in their second reviews, noting that all equation, reference and figure numbers refer to the main text unless otherwise stated. We have accordingly expanded our discussion of these issues in our current manuscript, where all recent revisions are marked in red.

Reply to Referee 1
Referee 1 asked us to clarify experimental consequences that distinguish our proposed two-phonon mechanism for quantum critical polar superconductivity from others previously proposed in the literature: "The authors proposed that their theory can be applied to many other materials systems (such as doped BaTiO3 and KTaO3, and polar metal LiOsO3). Experimentally superconductivity has been found in bulk KTaO3 and recently at KTaO3 (111) surface. My question is whether there is an experimental smoking gun (e.g. gap/Tc ratio) that can show that the observed superconductivity arises from the pairing mechanism proposed in the current theory, rather than from the conventional electron-phonon mechanism, or other related theories (such as  The scaling of T c with phonon frequency, discussed in the text surrounding Eq. 17, is another defining signature for c s k F ≪ ω T (n e ) and for c s k F ≫ ω T (n e ). An important point here is that T c is very sensitive to the TO phonon energy at the low doping concentrations we consider (see also Fig. 3).
Finally, in typical quantum critical polar pairing scenarios (e.g. Annals of Physics, Volume 417,168142 (2020)), the normal state close to the polar QCP is expected to be a non-Fermi liquid. This is not the case in the energy fluctuation scenario since the coupling in Eq. 1 is irrelevant in the RG sense.

Reply to Referee 2
Referee 2 asked us to provide additional clarification regarding the distinction between the "energy fluctuations exchange" and the "two-phonon mechanism": "I would only appreciate if the authors explain me once again in more detail the difference between "energy fluctuations exchange" and "two-phonon mechanism". I see the authors emphasize it several times in their reply that these are actually different, though I still don't quite understand why. I always thought (maybe mistakenly) that the term "two-phonon mechanism" simply implies the coupling between electron density and the square of the lattice polarization, In the long wavelength (small momentum q → 0) limit near a polar QCP, the scaling of the interaction u will lead to singular corrections; in this case individual phonons will not be well-defined as quasiparticles and the behavior of ⟨ ⃗ P 2 (x) ⃗ P 2 (x ′ )⟩ will be determined by the universality class of the transition. In the current work we assumed that the relevant momenta q ∼ k F are large enough to be beyond the critical region, where the singular critical corrections can be ignored. In 3D they are logarithmic, while in 2D phonon quasiparticles are more seriously affected by their interactions u leading to an anomalous dimension D(x − x ′ ) ∼ 1 |x−x ′ | d−1+η in the long-wavelength limit (see Supplementary Section 2). Therefore, in the absence of well-defined phonon quasiparticles, it is more appropriate to attribute the interaction to energy fluctuation exchange, as energy of the lattice fluctuations remains a well-defined concept even when phonons are not.
In our work we do focus on the carrier density regime where the electron-electron interaction can indeed be approximated well by the perturbative two-phonon exchange, since a weak-coupling approach is appropriate. Still we believe that our theory provides a first step towards the study of electron pairing due to non-analytical energy fluctuations, which then